Models of Economic Growth B
Outline:
1. Endogenous growth models
– Slides 1-22 (these slides offer a slightly
more detailed treatment of endogenous
models than Chapter 8)
2. The convergence debate
– Slides 23 – 34 (this is additional to
material covered in the book)
Endogenous growth models topics
• Recap on growth of technology (A) in Solow model
(…..does allow long run growth)
• Endogenous growth models
• Non-diminishing returns to ‘capital’
• Role of human capital
• Creative destruction models
• Competition and growth
• Scale effects on growth
Exogenous technology growth
• Solow (and Swan) models show that technological
change drives growth
• But growth of technology is not determined within the
model (it is exogenous)
• Note that it does not show that capital investment is
unimportant ( A  y and  MPk, hence k)
• In words …. better technology raises output, but also
creates new capital investment opportunities
• Endogenous growth models try to make endogenous
the driving force(s) of growth
• Can be technology or other factors like learning by
workers
The AK model
• The ‘AK model’ is sometimes termed an ‘endogenous
growth model’
• The model has Y = AK
where K can be thought of as some composite ‘capital
and labour’ input
• Clearly this has constant marginal product of
capital (MPk = dY/dK=A), hence long run growth is
possible
• Thus, the ‘AK model’ is a simple way of illustrating
endogenous growth concept
• However, it is very simple! ‘A’ is poorly defined, yet
critical to growth rate
• Also composite ‘K’ is unappealing
The AK model in a diagram
output per worker
y
Y=AK
Constant slope
represents constant
marginal product of
capital
Gross investment line
Depreciation line
Gap between lines
represents net investment,
which is always positive.
K
Endogenous technology growth
• Suppose that technology depends on past
investment (i.e. the process of investment generates
new ideas, knowledge and learning).
dA
A  g (K )
where
0
dK
Specifically, let A  K 
 0
Cobb-Douglas production function
Y  AK  L1  [ K  ]K  L1  K    L1
If + = 1 then marginal product of capital is
constant (dY/dK = L1- ).
• Assuming A=g(K) is Ken Arrow’s (1962)
learning-by-doing paper
• Intuition is that learning about technology
prevents marginal product declining
y=kL1-
output per worker
y
Slope = marginal
product = L1- =
constant (if labour
force constant)
sy
dk
Gap between lines
represents net
investment. Always
positive, hence growth
k
“Growth of capital”
Situation on growth diagram
sy
Distance between lines
represents growth in capital
per worker
d+n
capital per worker
k=K/L
Increasing returns to scale
Y K
   1
L
with     1
• “Problem” with Y = K1L1- is that it exhibits
increasing returns to scale (doubling K and L,
more than doubles Y)
• IRS  large firms dominate, no perfect
competition (no P=MC, no first welfare theorem,
…..)
• …. solution, assume feedback from investment
to A is external to firms (note this is positive
externality, or spillover, from microeconomics)
Knowledge externalities
A firm's production function is
 1
Yi  Ai K i Li
but Ai depends on aggregate capital
(hence firm does not 'control' increasing returns)
• Romer (1986) paper formally proves such a model
has a competitive equilibrium
• However, the importance of externalities in
knowledge (R&D, technology) long recognised
• Endogenous growth theory combines IRS,
knowledge externalities and competitive behaviour
in (dynamic optimising) models
More formal endogenous growth models
• Romer (1990), Jones (1995) and others use a model
of profit-seeking firms investing in R&D
• A firm’s R&D raises its profits, but also has a positive
externality on other firms’ R&D productivity (can have
competitive behaviour at firm-level, but IRS overall)
• Assume Y=K(ALY)1
• Labour used either to produce output (LY) or
technology (LA)
• As before, A is technology (also called ‘ideas’ or
‘knowledge’)
• Note total labour supply is L = LY + LA
Romer model
dA
Assume
 d LA A
d >0
dt
This is differential equation. Can A have constant growth rate?
Answer: depends on parameters  and  and growth of LA
Romer (1990) assumed:
  1,   1
dA
hence
 d LA A
dt
dA

/ A  d LA (>0 if some labour allocated to research)
dt
If A has positive growth, this will give long run growth in GDP p.w.
Note that there is a 'scale effect' from LA
Note ‘knife edge’ property of =1. If >1, growth rate will accelerate over
time; if <1, growth rate falls.
Jones model (semi-endogenous)
  0,   1 (Jones, 1995)
dA
dA
A d LA A d LA
 
Now
 d LA A

/ A 
= 1
dt
dt
A
A
A
Can only have positive long run growth if far right term is constant
LA
A
This only when 
 (1   )
LA
A
or
A
 LA

A (1   ) LA
In words: growth of technology = constant  labour growth
• No scale effects, no ‘knife edge’ property, but
requires (exogenous) labour force growth hence “semiendogenous” (see Jones (1999) for discussion)
Human capital – the Lucas model
• Lucas defines human capital as the skill
embodied in workers
• Constant number of workers in economy is N
• Each one has a human capital level of h
• Human capital can be used either to produce
output (proportion u)
• Or to accumulate new human capital
(proportion 1-u)
• Human capital grows at a constant rate
dh/dt = h(1-u)
Lucas model in detail
• The production of output (Y) is given by
Y = AKα (uhN)1-α ha g
where 0 <  < 1 and g  0
• Lucas assumed that technology (A) was constant
• Note the presence of the extra term hag - this is
defined as the ‘average human capital level’
• This allows for external effect of human capital that
can also influence other firms, e.g. higher average
skills allow workers to communicate better
• Main driver of growth - As h grows the effect is to
scale up the input of workers N, so raising output Y
and raising marginal product of capital K
Creative destruction and firm-level activity
• many endogenous growth models assume profitseeking firms invest in R&D (ideas, knowledge)
– Incentives: expected monopoly profits on new product or
process. This depends on probability of inventing and, if
successful, expected length of monopoly (strength of
intellectual property rights e.g. patents)
– Cost: expected labour cost (note that ‘cost’ depends on
productivity, which depends on extent of spillovers)
• models are ‘monopolistic competitive’ i.e. free entry into
R&D  zero profits (fixed cost of R&D=monopoly
profits). ‘Creative destruction’ since new inventions
destroy markets of (some) existing products.
• without ‘knowledge spillovers’ such firms run into
diminishing returns
• such models have three potential market failures,
which make policy implications unclear
Market failures in R&D growth models
1. Appropriability effect (monopoly profits of a new
innovation < consumer surplus)  too little R&D
2. Creative-destruction, or business stealing, effect
(new innovation destroys profits of existing firms),
which private innovator ignores  too much R&D
3. Knowledge spillover effect (each firm’s R&D helps
reduce costs of others innovations; positive
externality)  too little R&D
The overall outcome depends on parameters and
functional form of model
What do we learn from such models?
• Growth of technology via ‘knowledge spillovers’ vital
for economic growth
• Competitive profit-seeking firms can generate
investment & growth, but can be market failures
(‘social planner’ wants to invest more since spillovers
not part of private optimisation)
• Spillovers, clusters, networks, business-university links
all potentially vital
• But models too generalised to offer specific policy
guidance
Competition and growth
• Endogenous growth models imply greater
competition, lower profits, lower incentive to do
R&D and lower growth (R&D line shifts down)
• But this conflicts with economists’ basic belief
that competition is ‘good’!
• Theoretical solution
– Build models that have optimal ‘competition’
– Aghion-Howitt model describes three sector model
(“escape from competition” idea)
• Intuitive idea is that ‘monopolies’ don’t innovate
Do ‘scale effects’ exist
• Romer model implies countries that have more ‘labour’
in knowledge-sector (e.g. R&D) should grow faster
• Jones argues this not the case (since researchers in US
 5x (1950-90) but growth still 2% p.a.
• Hence, Jones claims his semi-endogenous model
better fits the ‘facts’, BUT
– measurement issues (formal R&D labs increasingly used)
– ‘scale effects’ occur via knowledge externalities (these may be
regional-, industry-, or network-specific)
– Kremer (1993) suggests higher population (scale) does
increase growth rates over last 1000+ years
• anyhow…. both models show  (the ‘knowledge
spillover’ parameter) is important
Questions for discussion
1. What is the ‘knife edge’ property of
endogenous growth models?
2. Is more competition good for economic
growth?
3. Do scale effects mean that China’s growth
rate will always be high?
References
Arrow, K. (1962). "The Economic Consequences of Learning by Doing." Review of
Economic Studies XXIX(80): 155-173.
Jones, C. (1995). "R&D Based Models of Economic Growth." Journal of Political
Economy 103(4): 759-784.
Jones, C. (1999) "Growth: With or Without Scale Effects?" American Economic Review
Papers and Proceedings, 89, 139-144.
Kremer, M. (1993). "Population Growth and Technological Change: One Million B.C. to
1990." Quarterly Journal of Economics 108: 681-716.
Lucas, R. E. (1988). "On the Mechanics of Economic Development." Journal of Monetary
Economics 22: 3-42.
Mankiw, N., D. Romer, et al. (1992). "A Contribution to the Empirics of Economic
Growth." Quarterly Journal of Economics 107(2): 407-437.
Romer, P. (1986). "Increasing Returns and Long Run Growth." Journal of Political
Economy 94(2): 1002-1037.
Romer, P. (1990). "Endogenous Technological Change." Journal of Political Economy
98(5): S71-S102.
Convergence debate
• Following slides discuss the above issue,
which is not included in detail in book
• However, it may be a topic to include in a
course on macroeconomics of growth, see:
– Milanovic, B. (2002). "Worlds Apart: Inter-National
and World Inequality 1950-2000". World Bank,
http://www.worldbank.org/research/inequality/worl
d%20income%20distribution/world%20apart.pdf.
– Pritchett, L. (1997). "Divergence, Big Time."
Journal of Economic Perspectives 11(3): 3-17.
– Stiglitz, J.-E. (2000). "Capital Market
Liberalization, Economic Growth, and Instability."
World Development 28(6): 1075-86.
Do poorer countries grow faster? (the
‘convergence’ debate)
Two common ways to assess convergence
1. Beta () convergence
2. Sigma (s) convergence
-convergence (use regression analysis)
growthi = constant +  (initial GDP p.w.)i
(i stands for a country. Test on sample of 60+)
If <0, poorer countries, on average, grow faster
s-convergence
measure dispersion (variance) of GDP per worker across
countries in a given year. If dispersion falls over time can
say countries ‘converging’.
8
-convergence, 110 countries, 1965-2000
Source: PWT 6.1
BWA
6
TWN
KOR
HKG
4
SGP
IRL
JPN
2
LUX
GRC
USA
0
ARG
-2
PER
MRT
AGO
NZL
CHE
VEN
NIC
-4
ZAR
0
10000
20000
30000
Real GDP per worker in 1965 (US$,PPP 96 prices)
Estimated  for above = 0.000. No beta convergence
40000
Problems and other evidence
• There are more than 110 countries (UN 191). The
poorest countries often don’t have data. Hence
above result could be mis-leading.
• L Pritchett (1997) “Divergence, Big Time”.
– 1870-1990, rich countries got much richer
– 9/1 ratio in 1870; 45/1 ratio in 1990
• Some view the 1960s-80s as good decades for
poorer countries – normally divergence
• “Conditional convergence”.
– If regression analysis controls for other factors (e.g.
investment), poorer countries do grow faster.
– Not very surprising …. what are other factors?
sigma (s) convergence
• Variance (110 countries GDP per worker) increases
over time  divergence since 60s
• But, if you weight countries according to population,
evidence shows convergence (this appears entirely
due to China, Milanovic, 2002, see next slide)
(note: if you do not weight you give China and, say,
Togo the same ‘importance’ or ‘weight’)
• Finally, researchers now working on ‘true’ world
inequality data (i.e. combine within country, and across
country, inequality). Initial results show ‘world’
inequality increasing since late 1980s
Gini measure of inter-country inequality
0.6
Gini index
World population-weighted
World unweighted without China and India
0.5
World unweighted
0.4
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
Year
Source: Data direct from Branko Milanovic, see Milanovic (2002). The higher
the Gini-coefficient, the more ‘unequal’ is GDP per capita across the world (see
endnotes)
2000
Twin peaks ?
.3
Some evidence that world income distribution is moving to ‘twinpeaks’ distribution…… but not very strong
1995 real GDP per worker
0
.1
Fraction
.1
.2
Fraction
.2
.3
1965 real GDP per worker
10000
20000
30000
40000
Real GDP per worker, US$ 1996 prices
0
0
0
10000
20000
30000
40000
50000
Real GDP per worker, US$ 1996 prices
60000
50000
60000
What are mechanisms driving
‘convergence’?
• Important to understand basic data, but real
issue is mechanisms
• Consider some ‘theory’ initially
– open economy growth models
– models of technological catch-up
• Note: this ‘convergence’ is not ‘Solow-Swan
convergence to steady state’
– can consider country convergence in S-S model
but must assume technology common to all
countries
Weighted world inter-country inequality (role of China and India)
0.600
World
0.560
World without
China
0.480
World without India and China
0.440
0.400
19
50
19
52
19
54
19
56
19
58
19
60
19
62
19
64
19
66
19
68
19
70
19
72
19
74
19
76
19
78
19
80
19
82
19
84
19
86
19
88
19
90
19
92
19
94
19
96
19
98
20
00
Gini coefficient
0.520
Summary (I)
Sigma (s) convergence
• Using unweighted measures, cross-country
evidence suggests ‘divergence’
• Weighted measures  convergence over
last 30 years due to performance of China
• However, most recent ‘world inequality’
measures based on within and across
country data,  divergence
Summary (II)
Beta () convergence
• No unconditional convergence
• There is conditional convergence (poorer
countries grow faster if you control for other
factors)
• Expect this (basic closed economy Solow
and endogenous growth models predict this)
Endnotes
Inequality/Convergence
• Gini coefficient, world income inequality.
– The Gini coefficient varies from 0 (perfect equality) to
1 (perfect inequality).
– The discussion related to this overlooks many
important and controversial issues. See Milanovic
(2002) for a discussion, or google it.
– Milanovic, B. (2002). "Worlds Apart: Inter-National
and World Inequality 1950-2000". World Bank,
http://www.worldbank.org/research/inequality/world%
20income%20distribution/world%20apart.pdf.